Optimal. Leaf size=50 \[ \frac{1}{972} \log \left (4 x^2+6 x+9\right )-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}} \]
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Rubi [A] time = 0.0481794, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1586, 2058, 618, 204, 628} \[ \frac{1}{972} \log \left (4 x^2+6 x+9\right )-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2058
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{3+2 x}{729-64 x^6} \, dx &=\int \frac{1}{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5} \, dx\\ &=\int \left (-\frac{1}{243 (-3+2 x)}+\frac{1}{54 \left (9-6 x+4 x^2\right )}+\frac{3+4 x}{486 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{1}{486} \log (3-2 x)+\frac{1}{486} \int \frac{3+4 x}{9+6 x+4 x^2} \, dx+\frac{1}{54} \int \frac{1}{9-6 x+4 x^2} \, dx\\ &=-\frac{1}{486} \log (3-2 x)+\frac{1}{972} \log \left (9+6 x+4 x^2\right )-\frac{1}{27} \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}}-\frac{1}{486} \log (3-2 x)+\frac{1}{972} \log \left (9+6 x+4 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0134169, size = 46, normalized size = 0.92 \[ \frac{1}{972} \left (\log \left (4 x^2+6 x+9\right )-2 \log (3-2 x)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 39, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( -3+2\,x \right ) }{486}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{972}}+{\frac{\sqrt{3}}{486}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3875, size = 51, normalized size = 1.02 \begin{align*} \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40525, size = 128, normalized size = 2.56 \begin{align*} \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.16509, size = 46, normalized size = 0.92 \begin{align*} - \frac{\log{\left (x - \frac{3}{2} \right )}}{486} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{972} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{486} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04905, size = 53, normalized size = 1.06 \begin{align*} \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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